Comparison of Numerical Solvers for Differential Equations for Holonomic Gradient Method in Statistics

TL;DR

This paper compares various numerical solvers for linear ODEs used in the holonomic gradient method, which is important for computing normalizing constants in statistical models.

Contribution

It provides a comparison of different numerical solvers for linear ODEs within the holonomic gradient method framework, highlighting their effectiveness.

Findings

Different solvers vary in accuracy and efficiency.

Some solvers are more suitable for specific types of holonomic systems.

The study guides the choice of numerical methods for statistical computations.

Abstract

Definite integrals with parameters of holonomic functions satisfy holonomic systems of linear partial differential equations. When we restrict parameters to a one dimensional curve, the system becomes a linear ordinary differential equation (ODE) with respect to a curve in the parameter space. We can evaluate the integral by solving the linear ODE numerically. This approach to evaluate numerically definite integrals is called the holonomic gradient method (HGM) and it is useful to evaluate several normalizing constants in statistics. We will discuss and compare methods to solve linear ODE's to evaluate normalizing constants.